Expert: Josh - 5/6/2009

Question
Hello:

If I want to invest $1000.00 in two accounts one at 2% and one at 5% for a total return of $44.00, I can determine the amounts for each account as follows:

2% of $1000.00 equals $20.00. $44.00 - $20 = $24.00
Both accounts are earning 2% and the other is earning 3% more. If I divide $24.00 by 3%, $24.00/3%, I get the partial answer of $800.00 for the 5% account. The 2% account has the amount of $1000.00 - $800.00 or $200.00.

My question is as follows: Can the above solution be used if three or more percentages are used for three or more separate accounts?

For example, divide $1000.00 in three accounts one earning 2%, one earning 3% and one earning 5% so that the total return is $44.00.

I'm only interested in this solution and if it can be used to determine the amounts for the three accounts and not some completed algebra solution.

I thank you for your reply.

Answer
Hello:

I have three comments in regard to your questions:

1. Your approach to the problem seems quite unusual at first glance. Even though you may not be aware of this, everything you have done is algebra in disguise. Because it is done using only numbers -- without the benefit of mathematical abstraction or variables -- it does not give you any insight.

A more proper way to set up this problem is as follows:
Let principal P=1000
   interest rates s=0.02, r=0.05
   interest earned at maturity I=44
   amount invested at s (2%), L=?

Then, the combined interest I=(P-L)*s+L*r, where P,s,r are known.
The amount you invest at interest rate r (2%) is given by:
L=(I-P*s)/(r-s) ....[1]

If you retrace your steps, you will find that you have used unwittingly the following relations:
y=P*s for the initial interest calculation (based on 2%), and
x=r-s for the difference in interest rate.

This fits in perfectly with the algebra construction.
From [1], we "x" and "y" as defined, L=(I-y)/x.

2. Although you may wish there were a simple recipe that does not involve algebra, all the steps you have taken (without knowing why they were taken) were in fact the exact footprints of the usual algebraic steps. In fact, getting rid of the symbols makes your approach rather ad-hoc and much more difficult to understand.

3. The short answer to your question for three accounts is NO, or perhaps, highly inefficient. To understand why, you will need to learn linear algebra to see that the system of equations is under-constrained when 3 accounts are involved. You will need to impose an extra condition, perhaps in relation to the proportion or how the money is split to obtain an unique solution.